**Cyclic Subgroups : **

If we pick some element **a** from a group *G* then we can consider the subset of all elements of *G* that are powers of **a**. This subset forms a subgroup of *G* and is called the **cyclic subgroup generated by a**. If forms a subgroup since it is

**Closed**. If you multiply powers of **a** you end up with powers of **a**
**Has the identity**. $a \ast a^{-1} = a^{0} = e$

For $\mathbb{Z_{30}}$( for additive operation),

$25+ 25= 50= 20 (\mod 30).$

$25+ 25+ 25= 75= 15 (\mod 30)$.

$25+ 25+ 25+ 25= 100= 10 (\mod 30)$.

$25+ 25+ 25+ 25+ 25= 125= 5 (\mod 30).$

$25+25+ 25+ 25+ 25+ 25= 150= 0 (\mod 50).$

$25+25+ 25+ 25+ 25+ 25+ 25= 175= 25 (\mod 30).$

There are $6$ such groups.

Is there any better approach ?